This invention relates to a radially wedging, frictional overrunning clutch having Z sprags of a profiled section (Z being the number of sprags) which are arranged in a circumferential series between a cylindrical inner track having a diameter D.sub.i =2R.sub.i and a cylindrical outer track which is arranged concentrically with respect to the inner cylindrical track and which has a diameter D.sub.a =2R.sub.a. The inner cylindrical track is formed by the outer cylinder face of an inner clutch ring whereas the outer track is formed by the inner cylinder face of an outer clutch ring. The sprags are disposed in such a manner that upon rotation of the clutch rings relative to one another in the locking direction, the sprags wedge with their clutch faces against the cylindrical tracks and generate, at the line of contact between sprag and inner track, a radially outwardly directed normal force N.sub.i in the inner ring and further generate at the line of contact between sprag and outer track, a radially inwardly directed normal force N.sub.a in the outer ring. The clutch faces of the sprags have at the line of contact with the inner or, as the case may be, the outer track, a radius of curvature r.sub.i and r.sub.a, respectively. The distance between the centers of the two curvatures is designated at c. The inner wedging angle .epsilon..sub.i and the outer wedging angle .epsilon..sub.a between the plane containing the two lines of contact and the plane containing one of these lines of contact and the rotary axis of the overrunning clutch are determined by the following equations: ##EQU1## AND, RESPECTIVELY, ##EQU2##
In this manner, the torques T.sub.i and T.sub.a related to the inner ring and the outer ring, respectively, are obtained as EQU T.sub.i = z N.sub.i R.sub.i tan .epsilon..sub.i ( 3)
and, respectively, EQU T.sub.a = z N.sub.a R.sub.a tan .epsilon..sub.a. (4)
Taking into consideration Hertz's law, for the normal forces one obtains ##EQU3## and ##EQU4## wherein .nu. is Poisson's number, E is the modulus of elasticity,
L.sub.i and L.sub.a are, respectively, the length of the sprag portion engaging the inner and the outer ring, PA1 p.sub.Hi and p.sub.Ha are, respectively, Hertz's stresses between the inner ring track and the inner clutch face and, respectively, between the outer ring track and the outer clutch face.
Further, it is the smaller value N of the two values N.sub.i and N.sub.a obtained for p.sub.H = p.sub. admissible which is to be used as the basis for computing the maximum transmissible torque.
The computation of sprag-type overrunning clutches, be it for the determination of the maximum transmissible torque at given outer dimensions or be it for determining structural dimensions for a given torque, is effected in a conventional manner with the aid of the above-given or related relationships. For a more detailed explanation of these relationships reference is made to FIG. 1 which shows a fragmentary radial section of an overrunning clutch with the more important forces appearing upon torque transmission at the sprag. The outer cylindrical surface of the inner ring 1 constitutes the inner sprag track 2, while the inner cylindrical surface of the outer ring 3 constitutes the outer sprag track 4. Between the inner ring and the outer ring there are positioned the circumferentially arranged sprags 5 which can wedge with their inner clutch face 6 against the inner track 2 and with their outer clutch face 7 against the outer track 4. Upon such an occurrence the forces illustrated in FIG. 1 are generated. In addition to the above-mentioned normal forces N.sub.i and N.sub.a there appear circumferential forces H.sub.i and H.sub.a. In order to ensure an equilibrium of force, the resultants of N.sub.i and H.sub.i and, respectively, N.sub.a and H.sub.a have to lie on the same line of action, must be oppositely oriented and must be of identical magnitude, as illustrated in FIG. 1. If one considers that EQU H.sub.i = N.sub.i tan.epsilon..sub.i
and, respectively, EQU H.sub.a = N.sub.a tan.epsilon..sub.a,
for the transmissible torque there can be obtained immediately the relationships (3) and (4) set forth earlier. It is noted that .epsilon..sub.i and .epsilon..sub.a are structurally predetermined magnitudes which may be constant or may have, in the wedging zone, a minimum value as disclosed in German Laid-Open Application (Offenlegungsschrift) No. 2,204,305 and German Pat. No. 1,199,066. The computation of .epsilon..sub.i and .epsilon..sub.a may be effected trigonometrically with the aid of equations (1) and (2), respectively. Since the wedging angles determine the ratio of the circumferential force to the normal force, one seeks to work, in the engaged (wedging) state, with possibly large wedging angles, that is, it is sought to generate with relatively small normal forces high circumferential forces and thus relatively high torques. The maximum of the wedging angle is determined by the frictional coefficient .mu..sub.o which depends on surface properties. The friction between two bodies, thus the torque transmission, is effected only as long as EQU H.sub.i .ltoreq. .mu..sub.o N.sub.i and H.sub.a .ltoreq. .mu..sub.o N.sub.a,
or, stated differently, as long as tan.epsilon..sub.i .ltoreq. .mu..sub.o and tan.epsilon..sub.a .ltoreq. .mu..sub.o.
Further, it has to be taken into account that during the engaging (coupling) step significantly smaller wedging angle values are necessary because first an oil film, usually coating all components, has to be penetrated before favorable frictional relationships set in which permit a relatively high wedging angle.
Upon the determination of normal forces an upper limit has to be observed as well. This upper limit is determined by the maximum admissible Hertz stress p.sub.H which is in the order magnitude of 400 kp/mm.sup.2, as long as the hardened ring material has a surface hardness of at least 60 HRc and a hardened depth of at least 0.8 mm and further, the sprags are formed of roller bearing steel hardened through the entire cross section.
A further limit value for the load bearing capacity of the overrunning clutch is the tangential tension .sigma..sub..sub..phi. a in the outer ring track and .sigma..sub..phi..sub.i in the inner ring track. During load, they must not be greater than the admissible tension .sigma..sub.adm. The permissible tension .sigma..sub.adm is obtained, after selection of a suitable safety factor, from the elastic limit .sigma..sub.s of the ring material utilized. The admissible tension is, for the usual materials in overrunning clutches, 60 kp/mm.sup.2.
With a given Poisson's number .nu. which is approximately 0.3 for steel and the modulus of elasticity E which is 21,000 kp/mm.sup.2 for steel, all magnitudes for determining the maximum permissible normal forces as well as the maximum transmissible torques for a given structure are known.
The outer dimensions, that is, the thickness and width of the inner and outer rings are not parameters in the presently used calculating method. It is known nevertheless that Hertz's stress is not always the load limit but that the load limit may also be caused by stresses in the rings and by elastic deformation of all other components. By virtue of elastic deformations a whole number of important characteristic magnitudes of the overrunning clutch are changed. The most sensitive variable here is the wedging angle. Its magnitude may vary by a factor of 2 in response to deformations. In case the wedging angle increases in an excessive manner with the elastic deformations, the frictional limit is reached with too small loads. As a result, the overrunning clutch will slip before a satisfactory material economy is achieved. On the other hand, for high torque transmissibility it is required that the wedge angle increase with increasing loads because the greater the wedge angle at a given normal force, the greater the circumferential force and thus the transmissible torque. In the work by C. B. Biezeno and R. Grammel, entitled TECHNISCHE DYNAMIK, volume 1, 2nd edition (publisher: Springer, 1953), there is given a method for computing the elastic deformations. This computing method, however, applies only in case of small wall thicknesses. It has been shown in practice, however, that sprag-type overrunning clutches require rings of substantial wall thicknesses. This computing method is therefore not adapted to determine the optimal dimensions of an overrunning clutch for transmitting the highest possible torque for a given available space of installation.